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| '''Indexed languages''' are a class of [[formal language]]s discovered by [[Alfred Aho]];<ref name="aho1968">{{cite journal | last = [[Alfred Aho|Aho]] | first = Alfred | year = 1968 | title = Indexed grammars—an extension of context-free grammars | journal = [[Journal of the ACM]] | volume = 15 | issue = 4 | pages = 647–671 | url = http://portal.acm.org/ft_gateway.cfm?id=321488&type=pdf&coll=GUIDE&dl=GUIDE,&CFID=17841194&CFTOKEN=70113868 | doi = 10.1145/321479.321488 }}</ref> they are described by [[indexed grammar]]s and can be recognized by [[nested stack automaton]]s.<ref name="partee_etal_1990">{{cite book | last = [[Barbara Partee|Partee]] | first = Barbara | coauthors = Alice ter Meulen, and Robert E. Wall | title = Mathematical Methods in Linguistics | year = 1990 | publisher = Kluwer Academic Publishers | pages = 536–542 | isbn = 978-90-277-2245-4 }}</ref>
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| Indexed languages are a proper [[subset]] of [[context-sensitive language]]s. They qualify as an [[abstract family of languages]] (furthermore a full AFL) and hence satisfy many closure properties. However, they are not closed under intersection or complement.<ref name="aho1968" /> Gerald Gazdar has characterized the mildly context-sensitive languages in terms of linear indexed grammars.<ref name="gazdar1998">{{cite book | chapter=Applicability of Indexed Grammars to Natural Languages | year=1988 | last=Gazdar | first=Gerald | title=Natural Language Parsing and Linguistic Theories | editor=U. Reyle and C. Rohrer | pages=69–94}}</ref>
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| The class of indexed languages has practical importance in [[natural language processing]] as a computationally affordable generalization of [[context-free languages]], since [[indexed grammar]]s can describe many of the nonlocal constraints occurring in natural languages.
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| ==Examples==
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| The following languages are indexed, but are not [[context-free language|context-free]]:
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| :<math> \{a^n b^n c^n d^n| n \geq 1 \} </math> <ref name="gazdar1998"/>
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| :<math> \{a^n b^m c^n d^m | m,n \geq 0 \}</math> <ref name="partee_etal_1990"/>
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| These two languages are also indexed, but are not even [[mildly context-sensitive language|mildly context sensitive]] under Gazdar's characterization:
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| :<math> \{a^{2^{n}} | n \geq 0 \}</math> <ref name="partee_etal_1990"/>
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| :<math> \{www | w \in \{a,b\}^+ \}</math> <ref name="gazdar1998"/>
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| On the other hand, the following language is not indexed:<ref name="Gilman.1996">{{cite journal| last = Gilman | first = Robert H. | title=A Shrinking Lemma for Indexed Languages| journal=[[Theoretical Computer Science (journal)|Theoretical Computer Science]]| year=1996| volume=163| | issue = 1–2 pages=277-281| url=http://www.sciencedirect.com/science/article/pii/0304397596002447/pdf?md5=33e62c7398fd8c6fb98f6759b154dcbd&pid=1-s2.0-0304397596002447-main.pdf | doi = 10.1016/0304-3975(96)00244-7 }}</ref>
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| :<math>\{(a b^n)^n | n \geq 0 \}</math>
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| ==Properties==
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| [[John Hopcroft|Hopcroft]] and [[Jeffrey Ullman|Ullman]] tend to consider indexed languages as a "natural" class, since they are generated by several formalisms, viz.{{#tag:ref|Introduction to automata theory, languages, and computation,<ref name="hopcroft_ullman_1979"> {{cite book | last = [[John Hopcroft|Hopcroft]] | first = John | coauthors = [[Jeffrey Ullman]] | title = [[Introduction to automata theory, languages, and computation]] | year = 1979 | publisher = Addison-Wesley | isbn = 0-201-02988-X | page = 390}} </ref>Bibliographic notes, p.394-395}}
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| * [[Alfred Aho|Aho]]'s [[indexed grammar]]s<ref name="aho1968"/>
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| * [[Alfred Aho|Aho]]'s one-way [[nested stack automata]]<ref>{{cite journal| author=Alfred Aho| title=Nested Stack Automata| journal=Journal of the ACM| year=1969| volume=16| number=3| pages=383-406}}</ref>
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| * [[Michael J. Fischer|Fischer]]'s macro grammars<ref>{{cite book| author=Michael J. Fischer| chapter=Grammars with Macro-Like Productions| title=Proc. 9th Ann. IEEE Symp. on Switching and Automata Theory (SWAT)| year=1968| pages=131-142}}</ref>
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| * [[Sheila Greibach|Greibach]]'s automata with stacks of stacks<ref>{{cite journal| author=Sheila A. Greibach| title=Full AFL's and Nested Iterated Substitution| journal=Information and Control| year=1970| volume=16| number=1| pages=7–35| url=http://www.sciencedirect.com/science/article/pii/S0019995870800390/pdf?md5=c960de360ba704b0f5403b6f95dd19f8&pid=1-s2.0-S0019995870800390-main.pdf}}</ref>
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| * [[Tom Maibaum|Maibaum]]'s algebraic characterization<ref>{{cite journal| author=T.S.E. Maibaum| title=A Generalized Approach to Formal Languages| journal=J. Computer and System Sciences| year=1974| volume=8| number=3| pages=409-439| url=http://www.sciencedirect.com/science/article/pii/S0022000074800310/pdf?md5=0c71b0ce688b0d4ca81a181089b02af9&pid=1-s2.0-S0022000074800310-main.pdf}}</ref>
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| Hayashi<ref>{{cite journal| author=T. Hayashi| title=On Derivation Trees of Indexed Grammars - An Extension of the ''uvwxyz'' Theorem| journal=Publication of the Research Institute for Mathematical Sciences| year=1973| volume=9| number=1| pages=61-92| institution=Research Institute for Mathematical Sciences| url=http://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=9&iss=1&rank=3}}</ref> generalized the [[Pumping lemma for context-free languages|pumping lemma]] to indexed grammars.
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| Conversely, Gilman<ref name="Gilman.1996"/><ref>{{cite journal| author=Robert H. Gilman| title=A Shrinking Lemma for Indexed Languages| year=1995| month=Sep| url=http://arxiv.org/abs/math/9509205}}</ref> gives a "shrinking lemma" for indexed languages.
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| ==See also==
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| * [[Chomsky hierarchy]]
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| * [[Indexed grammar]]
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| * [[Mildly context-sensitive language]]
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| ==References==
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| <references/>
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| ==External links==
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| * [http://www.cogs.susx.ac.uk/research/nlp/gazdar/nlp-in-prolog/ch04/chapter-04-sh-1.6.3.html#sh-1.6.3 "NLP in Prolog" chapter on indexed grammars and languages]
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| {{Formal languages and grammars}}
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| {{DEFAULTSORT:Indexed Language}}
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| [[Category:Formal languages]]
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| {{comp-sci-theory-stub}}
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